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IMO ShortList 2008, Number Theory problem 4

Source: IMO ShortList 2008, Number Theory problem 4

July 9, 2009
number theorybinomial coefficientsmodular arithmeticIMO ShortlistHi

Problem Statement

Let n n be a positive integer. Show that the numbers \binom{2^n \minus{} 1}{0},\; \binom{2^n \minus{} 1}{1},\; \binom{2^n \minus{} 1}{2},\; \ldots,\; \binom{2^n \minus{} 1}{2^{n \minus{} 1} \minus{} 1} are congruent modulo 2n 2^n to 1 1, 3 3, 5 5, … \ldots, 2^n \minus{} 1 in some order. Proposed by Duskan Dukic, Serbia
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